On the Shabbat before Rosh Chodesh, the custom is to announce the day or days on which the new month will fall, and to pray that “HaKadosh Baruch Hu renew it for us and all of Israel for goodness and blessing.” In a certain way, this also commemorates the sanctification of the new moon that the Beit Din used to perform. Therefore, the custom is to announce the exact time at which the new moon will appear (the molad). This is also why we stand during the ceremony, because the people used to stand before the Beit Din when it performed the mitzvah of Kiddush HaChodesh. There is only one Rosh Chodesh before which we do not bless the new month – Tishrei, because Rosh Chodesh Tishrei is also the holiday of Rosh HaShanah, and everyone knows that it’s coming.
We announce the coming of the new month specifically on Shabbat because that is when the entire congregation is gathered in the synagogue, and everyone will hear when Rosh Chodesh is slated to fall. Furthermore, all the days of the week receive blessing from Shabbat, and even the sanctity of Rosh Chodesh flows from the Shabbat that precedes it. Therefore, we bless the new month on that day. This is why we begin to feel on that Shabbat the festiveness of the forthcoming Rosh Chodesh.
Mishna Berura 417:1; Igrot Moshe, Orach Chaim 1:142; Sefer Rosh Chodesh 1:1-9. In notes 4-8 [of the last source], the author quotes the Yera’im, the Shibolei HaLeket, and the Or Zaru’a who say that the reason for Birkat HaChodesh is in order to let people know when Rosh Chodesh will be. The Ra’avyah writes that it is in commemoration of the Kiddush HaChodesh ceremony. For this reason, some communities give the Rabbi the honor of blessing the new moon (Sefer Rosh Chodesh 1:7). What I wrote concerning Tishrei is found in Sha’ar HaTziyun 417:2. For alternative reasons, see Sefer Rosh Chodesh 1:25. Some communities do not bless the month of Av, because of the calamities that occurred during that month, but the prevalent custom is to bless it. So explains Rav Kook, in Olat Re’iyah, vol. 2, p. 121, and Sefer Rosh Chodesh 1:27.
 This is why Ashkenazim refrain from saying memorial prayers for the dead on Shabbat Mevarchim (Rama, Orach Chaim 284:7, Mishna Berura 284:17). They even omit the Av HaRachamim prayer, which is said in memory of those killed in sanctification of God’s name [during the Crusades]. The only exceptions are the two Sabbaths prior to Rosh ChodeshIyar and Sivan, because the holy martyrs were killed [mainly] during these two months. My teacher and rabbi, Rav Tzvi Yehudah HaKohen Kook, used to recite Av HaRachamim even on [an ordinary] Shabbat Mevarchim. He held that after the Holocaust it should be said on every Shabbat, just like it is said before Rosh ChodeshIyar and Sivan.